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I mostly wanted to record the correct meaning of this term. Then maybe later I can use this as a reference to fix Wikipedia (^_^). But there's a bit more here too.
Well, definitions are definitions, neither correct nor incorrect except in some “moral” sense (using the term somewhat loosely, but see here for example). I’m not convinced it makes a much difference whether you take the imaginary part of to b or , but the former is so standard that I think it would be unwise to mess with it (since the advantages if any would be pretty small).
Thanks for the reference to Eugenia Cheng’s abstract: I wasn’t aware of her talk.
A possible ethical reason for defining the imaginary part of as would be that taking the real part is a projection of the 1-dimensional complex vector space onto a 1-dimensional real vector subspace. As such, taking the imaginary part should morally also be a projection onto a 1-dimensional real vector subspace. These two projections are part of the data which describes the 1-dimensional complex vector space C as a direct sum of two 1-dimensional real vector subspaces such that the conjugation involutive endomorphism of C is the direct sum of two involutive endomorphisms, one identity and the other being inversion. Furthermore, the almost complex structure on C is the direct sum of two morphisms between these direct summands.
That imaginary subspace being isomorphic to , I don’ t see how it makes any real difference. Any structure on one transfers across the isomorphism to a structure on the other.
The isomorphism allows you to transport a structure from one to the other. This will be my last comment on this topic.
In my experience, is totally standard.
On my understanding, the usual meaning of ‘imaginary part’ is as Todd and Mike said, but actually I think that Colin's is the better (more moral) definition. The usual way relies on the existence of an isomorphism between the real line and the purely imaginary line, but it is only our great good fortune that this isomorphism exists when working with complex numbers. (And even so, we're not as fortunate as we should be, since we have to arbitrarily pick one of equally canonical ones.) But when working with any higher-dimensional kind of hypercomplex numbers (such as the quaternions), we don't even have that option!1 So we really should take the imaginary part of a complex number to be purely imaginary, contrary to the usual definition, just as we do with quaternions. (And that is what people normally do with quaternions; they have to.)
But that still causes us no problems with terminology: The imaginary part of any hypercomplex number may not be an imaginary number (it usually is, but not when it's ), but it is a purely imaginary number.
The adjective ‘purely’ here is an example of a sort of red herring that one might call a degenerate red herring: it is only a red herring in degenerate cases. Another example is ‘homogenous’ in ‘homogeneous polynomial of degree ’.
Actually, the real numbers are themselves a kind of hypercomplex number in a trivial way, and the imaginary part of any of those is zero, any way you look at it. This works because what we actually need is not an isomorphism from purely imaginary numbers to real numbers but an embedding. Still, the only kinds of hypercomplex numbers where we have such a thing are those of dimension at most . ↩
I think that it is the same reason that we say “irrational” instead of “not rational”. For that matter, it is the same reason that we say “inhabited” instead of “not empty”. You seem to have drawn an analogy imaginary : real :: empty : inhabited, but the better analogy is imaginary : real :: inhabited : empty. There are many ways to be imaginary (many positive and many negative imaginary parts) but only one way to be real (one zero imaginary part); there are many ways to be inhabited (many positive cardinalities), but only one way to be empty (one zero cardinality). This is particularly clear in constructive mathematics; a complex number is real if it is not imaginary, but the contrapositive is invalid; a set is empty if it is not inhabited, but the contrapositive is invalid. There is a long tradition in constructive mathematics of inventing words for concepts that are classically negations of familiar concepts, so if we did not historically have the word “imaginary”, then the constructive analysts would have invented it!
A set is inhabited if there exists an element (over this set). A complex number is real if it is invariant under conjugation. A real number is rational if it is a quotient of natural numbers. On the other hand, it seems that empty, imginary and irrational are defined by inequalities.
So it seems that I believe in the following analogy that inhabited: real: rational :: empty, imaginary :irrational.
Could you explain further how does having many or one way to define something is related to defining something by an inequality?
A set is inhabited if its cardinality is positive; this is an inequality. More generally, there are typically many ways that things can be different (any property that distinguishes them renders them different) but only one way to be equal (every property must agree, pace Leibniz). This is only heuristic, of course.
Might we say here that affirmable statements in constructive logic typically correspond to open set conditions? See for example this MO discussion, with extensive commentary by Toby. Thus “imaginary” is an affirmable condition; “real” is not.
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